Some Remarks on Vector Potentials for Maxwell's Equations in Space-Time Carnot Groups

Annalisa Baldi; Bruno Franchi

Bollettino dell'Unione Matematica Italiana (2012)

  • Volume: 5, Issue: 2, page 337-355
  • ISSN: 0392-4041

Abstract

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In this paper we prove a Γ -convergence result for time-depending variational functionals in a space-time Carnot group ℝ × 𝔾 arising in the study of Maxwell's equations in the group. Indeed, a Carnot groups 𝔾 (a connected simply connected nilpotent stratified Lie group) can be endowed with a complex of ``intrinsic'' differential forms that provide the natural setting for a class of ``intrinsic'' Maxwell's equations. Our main results states precisely that a the vector potentials of a solution of Maxwell's equation in ℝ × 𝔾 is a critical point of a suitable functional that is in turn a Γ -limit of a sequence of analogous Riemannian functionals.

How to cite

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Baldi, Annalisa, and Franchi, Bruno. "Some Remarks on Vector Potentials for Maxwell's Equations in Space-Time Carnot Groups." Bollettino dell'Unione Matematica Italiana 5.2 (2012): 337-355. <http://eudml.org/doc/290932>.

@article{Baldi2012,
abstract = {In this paper we prove a $\Gamma$-convergence result for time-depending variational functionals in a space-time Carnot group $\mathbb\{R\} \times \mathbb\{G\}$ arising in the study of Maxwell's equations in the group. Indeed, a Carnot groups $\mathbb\{G\}$ (a connected simply connected nilpotent stratified Lie group) can be endowed with a complex of ``intrinsic'' differential forms that provide the natural setting for a class of ``intrinsic'' Maxwell's equations. Our main results states precisely that a the vector potentials of a solution of Maxwell's equation in $\mathbb\{R\} \times \mathbb\{G\}$ is a critical point of a suitable functional that is in turn a $\Gamma$-limit of a sequence of analogous Riemannian functionals.},
author = {Baldi, Annalisa, Franchi, Bruno},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {337-355},
publisher = {Unione Matematica Italiana},
title = {Some Remarks on Vector Potentials for Maxwell's Equations in Space-Time Carnot Groups},
url = {http://eudml.org/doc/290932},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Baldi, Annalisa
AU - Franchi, Bruno
TI - Some Remarks on Vector Potentials for Maxwell's Equations in Space-Time Carnot Groups
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/6//
PB - Unione Matematica Italiana
VL - 5
IS - 2
SP - 337
EP - 355
AB - In this paper we prove a $\Gamma$-convergence result for time-depending variational functionals in a space-time Carnot group $\mathbb{R} \times \mathbb{G}$ arising in the study of Maxwell's equations in the group. Indeed, a Carnot groups $\mathbb{G}$ (a connected simply connected nilpotent stratified Lie group) can be endowed with a complex of ``intrinsic'' differential forms that provide the natural setting for a class of ``intrinsic'' Maxwell's equations. Our main results states precisely that a the vector potentials of a solution of Maxwell's equation in $\mathbb{R} \times \mathbb{G}$ is a critical point of a suitable functional that is in turn a $\Gamma$-limit of a sequence of analogous Riemannian functionals.
LA - eng
UR - http://eudml.org/doc/290932
ER -

References

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